The problem of searching for gapped repeats and palindromes in words is not so new (see [1]}, [2]}, [3]}), and different solutions were proposed depending on the type of restrictions imposed on the gap.
In [4]}, Kolpakov and Kucherov introduced the notion of \(\alpha \) -gapped palindromes, and showed how to compute
the set \({\mathcal {G }}^{{\intercal }}_\alpha (w)\)  of all maximal \(\alpha \) -gapped palindromes in \({\mathcal {O}}(\alpha ^2 n + |{\mathcal {G }}^{{\intercal }}_\alpha (w)|)\)  time for any input word \(w\)  of length \(n\)  over constant alphabets.
In [5]}, Kolpakov et al. introduced the notion of \(\alpha \) -gapped repeats, and showed that the number of maximal \(\alpha \) -gapped repeats in a word \(w\)  of length \(n\)  is \({\mathcal {O}}(\alpha ^2 n)\) , and that this maximal set \(\mathcal {G }_\alpha (w)\)  can be computed in \({\mathcal {O}}(\alpha ^2 n)\)  time.
They suggested two open problems concerning this:
